We present an algorithm that constructively produces a solution to the k-DOMINATING SET problem for planar graphs in time O(c √ k n), where c = 4 6 √ 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O( √ γ (G)), and that such a tree decomposition can be found in O( √ γ (G)n) time. The same technique can be used to show that the k-FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n) time, where c 1 = 3 36 √ 34and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-DOMINATING SET, e.g., k-INDEPENDENT DOMINATING SET and k-WEIGHTED DOMINATING SET.

Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving N P-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = N P, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363-384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.Key words. parameterized algorithm, planar graph, dominating set, vertex cover, independent set, kernel AMS subject classifications. 05C85, 68Q17 DOI. 10.1137/050646354 1. Introduction. Many practical algorithms for N P-hard problems start by applying data reduction subroutines to the input instances of the problem. The hope is that after the data reduction phase the instance of the problem has shrunk to a moderate size. This makes the applicability of a second phase, such as a branchand-bound phase, to the resulting instance more feasible. Weihe showed in [41] how a practical preprocessing algorithm for a variation of the dominating set problem, called the red/blue dominating set problem, resulted in breaking up input instances of the problem into much smaller instances. Langston, and Shanbhag [2], in their implementation of algorithms for the vertex cover problem,

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We analyze edge dominating set from a parameterized perspective. More specifically, we prove that this problem is in FPT for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up the decision algorithm.

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8 k n) and O(8 k k + n 3 ), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general "annotated" problem on black/white graphs that asks for a set of k vertices (black or white) ଁ An extended abstract of this paper appeared under nearly the same title in that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.

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The k-LEAF OUT-BRANCHING problem is to find an out-branching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the k-LEAF-OUT-BRANCHING problem. We give the first polynomial kernel for ROOTED k-LEAF-OUT-BRANCHING, a variant of k-LEAF-OUT-BRANCHING where the root of the tree searched for is also a part of the input. Our kernel with O(k 3 ) vertices is obtained using extremal combinatorics.For the k-LEAF-OUT-BRANCHING problem, we show that no polynomial-sized kernel is possible unless coN P is in N P/ poly. However, our positive results for ROOTED k-LEAF-OUT-BRANCHING immediately imply that the seemingly intractable k-LEAF-OUT-BRANCHING problem admits a data reduction to n independent polynomialsized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem regarding "cheat kernelization" raised by Mike Fellows and Jiong Guo independently.

We consider the following problem (and variants thereof) that has important applications in the construction and evaluation of phylogenetic trees: Two rooted unordered binary trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a one-to-one correspondence between pairs of leaves of the different trees. Our goal is to find two planar embeddings of the two trees (drawn without crossings) that minimize the number of crossings of the matching edges. We derive both (classical) complexity results and (parameterized) algorithms for this problem (and some variants thereof). 1